So how does the router know it is out of range vs asleep?
For a wireless connection, the router usually DOES know this -- your phone (or other wireless device) generally goes into a special lower-power mode where in the wireless does not just "shut off" as you are thinking. Your phone will issue a special sleep command to the router telling it "I am going to low-power sleep mode". When it awakes to check traffic it does not have to fully reestablish a connection like a full break (i.e. does not to a full negotiation again). Routers which support sleep mode can queue up some broadcasts destined for the sleeping device and send them upon wakeup.
Unfortunately, that right there just pretty much exhausted my knowledge of the sleep status. It is, however, a part of the 802.11 standard, so a google search could bring up far more detail than anyone is likely willing to read (sorry electrical engineers out there, nobody else cares that much as long as it works!
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A
total guess, but the router may know the device is in this sleep state, but I am guessing FING and the other protocols outside the router are working a level or two up the networking stack, so there would be no requirement that the phone check in to these services when it awakes to see if it needs to process traffic.
Adam